Using a Fermionic Ensemble of Systems to Determine Excited States

TL;DR

This paper introduces a novel numerical approach using a fermionic ensemble of quantum systems to efficiently compute excited states, leveraging the exclusion principle and a generalized Feynman-Kac formula.

Contribution

The paper presents a new method that generalizes the Feynman-Kac formula by employing a fermionic ensemble to determine excited states of quantum systems.

Findings

Successfully applied to a 1D oscillator

Effective for a chain of coupled particles

Enables sampling of multiple low-energy states

Abstract

We discuss a new numerical method for the determination of excited states of a quantum system using a generalization of the Feynman-Kac formula. The method relies on introducing an ensemble of non-interacting identical systems with a fermionic statistics imposed on the systems as a whole, and on determining the ground state of this fermionic ensemble by taking the large time limit of the Euclidean kernel. Due to the exclusion principle, the ground state of an $n$-system ensemble is realized by the set of individual systems occupying successively the $n$ lowest states, all of which can therefore be sampled in this way. To demonstrate how the method works, we consider a one-dimensional oscillator and a chain of harmonically coupled particles.